So
Xx times Xy = (-xf'(x) cos y, -xf'(x) sin y, x).
If x 0 then X is regular, and for
x > 0 the unif normal is

so again a modified Gauss mapping, equivalent to N is given by

The parabolic set occurs when
0 = = -f'(x) f"(x),
i.e. at extrema or at inflections of the profile curve. Furthermore,
grad = (-(f"(x))2 - f'(x) f'''(x), 0),
so Ñ is good if f"(x) = 0 implies
f'(x) 0 and
f'''(x) 0.

If x0 is a value for which
f''(x0) = 0, then the parabolic
curve can be parametrized by
x(t) = x0,
y(t) = t, and we obtain

If f'(x0) 0 and
f'''(x0) 0, then the Gauss map
has an ordinary fold along the parabolic curve. For example, these
conditions are satisfied by the bell surface:

On the other hand, if x0 is a value for which
f'(x0) = 0 and
f"(x0) 0, then the Gauss map
is good, but not excellent, because the parabolic curve is
parametrized by x(t) = x0,
y(t) = t, and
N(t) = (0, 0) for all t. An example is the top
of a torus of revolution:

so N' = 0 if and only if = 0 and
then N" = 's'P. So the Gauss
map N is excellent if ' 0 whenever
= 0, and then N(x, y) has a
cusp at (t ,±/2) if and only if
(t) = 0.

For example consider the warped torus, a canal surface of
the space curve

(t) = (cos t, sin t, sin(nt))

where n is an integer, n 2. The
curvature of is nowhere zero, since
(s')3 = |(' x")| 1.
Furthermore

so = 0 if and only if
t = /2n, 3/2n, ...,
(4n - 1)/2n, provided that
0. Taking derivatives of
2(s')^6 and
n(1 - n2)cos nt
shows that tau = 0 implies ' 0, so
long as 0. Therfore a canal surface of has
an excellent Gauss map, with 4n cusps. For = 0 a
canal surface of is a torus of revolution, and each component
of the parabolic curve is collapsed to a point by the Gauss map.

Figure 2.15

The warped torus (n = 2).

Figure 2.16

The spherical image of the warped torus
(n = 2). (The parabolic image is two anitpodal
curves with 4 cusps each. The left picture is the spherical image of
the hyperbolic region, the right is the spherical image of the
elliptic region.)

Figure 2.17

The warped torus (n = 3).

Figure 2.18

The spherical image of the warped torus
(n = 3). (The parabolic image is two antipodal curves
with 3 cusps each. These curves are doubly covered by the parabolic
curve itself. The left picture is the spherical image of the
hyperbolic region, the right is the spherical image of the elliptic
region.)